Genuine $k$-partite correlations and entanglement in the ground state of the Dicke model for interacting qubits

TL;DR

This work analyzes genuine multipartite correlations in the generalized Dicke model with qubit-qubit interaction, using GMC to quantify all genuine $k$-partite correlations across partitions and to identify their distribution among subsystems. It demonstrates that GMC signals both first- and second-order quantum phase transitions, and uses Quantum Fisher Information to witness genuine $k$-partite entanglement, complemented by generalized global entanglement for pure ground states. The study provides detailed numerical results for a five-qubit system, revealing how competition between qubit-qubit and qubit-cavity couplings shapes multipartite correlations, and discusses an experimentally viable mapping to NV centers coupled to magnons in solids. Overall, the results illuminate the structure of classical and quantum correlations in many-body Dicke-type systems and propose a solid-state platform for realizing and probing these phenomena.

Abstract

Here, we calculate and study correlations of the Dicke model in the presence of qubit-qubit interaction. Whereas the analysis of correlations among its subsystems is essential for the understanding of corresponding critical phenomena and for performing quantum information tasks, the majority of correlation measures are restricted to bipartitions due to the inherent challenges associated with handling multiple partitions. To circunvent this we employ the calculation of Genuine Multipartite Correlations (GMC) based on the invariance of our model under particle permutation. We then quantify the correlations within each subpart of the system, as well as the percentage contribution of each GMC of order $k$, highlighting the many-body behaviors for different regimes of parameters. Additionally, we show that GMC signal both first- and second-order quantum phase transitions present in the model. Furthermore, as GMC encompasses both classical and quantum correlations, we employ Quantum Fisher Information (QFI) to detect genuine multipartite entanglement. Ultimately, we map the Dicke model with interacting qubits to spin in solids interacting with a quantum field of magnons, thus demonstrating a potential experimental realization of this model.

Figures (3)

• Figure 1: GMC of order $k$, $I^k$, for the ground state of interacting qubits resonant with the cavity mode, $\omega_0=\omega_c=1\xout{.0}$. Considering $N=5$ qubits inside the cavity, in a) we plot $I^k$ as function of interqubit coupling $\eta/N$ for $\lambda = 0$, $0.75$, and $1.79$, while in b) we plot $I^k$ as function of qubit-cavity coupling strength $\lambda/\sqrt{N}$ for $\eta = 0$, $1.5$, and $3$. The GMC are able of distinguishing both the first-order quantum phase transition (QPT), as pictured in a) for $\lambda=0$, as well as the transition from the normal to the superradiant phases, shown in b) for $\eta=0$. In c), we plot the GMC as function of both $\eta/N$ and $\lambda/\sqrt{N}$ for the ground state of $N=5$ qubits with $\omega_0=\omega_c=1$ and $k=1,2,4,5$.
• Figure 2: Maximum value of the QFI and generalized global entanglement for the ground state of the generalized Dicke model with $N=5$ qubits interacting resonantly with the cavity, $\omega_0=\omega_c =1$. (a) The $F_{max}/N$ for the whole range of parameters $\eta$ and $\lambda$. The discontinuities along the axis $\eta/N$ show the first-order QPTs, while the deep along the axis $\lambda/\sqrt{N}$ indicates the second-order QPT. (b) We observe the step-like behavior of the generalized global entanglement in the ground state of the system for $\ell=1,2,3,4$ as a function of $\eta/N$ for $\lambda=0$.
• Figure 3: (a) Plot of the GMC $I^k$ as a function of interqubit coupling $\eta$ for the ground state of $N=20$ qubits resonant with the cavity mode $(\omega_0=\omega_c=1.0)$ with $\lambda=0$. The first-order QPTs are all captured by the GMC, even though for small values of genuine $k$-partite correlations the steps become less evident, as noticed in the inset. (b) The plot of the total correlations $I^1$ as a function of the qubit-cavity coupling $\lambda$ for $\eta=0$ and increasing values of the number of qubits, $N$. The second-order QPT, which happens in the thermodynamic limit $N\rightarrow\infty$, is captured by the abrupt increase of total correlations between the normal and superradiant phase at $\lambda_c/\sqrt{N}=1/2$.